This book is an introduction to pattern theory, the theory behind the task of analyzing types of signals that the real world presents to us. It deals with generating mathematical models of the patterns in those signals and algorithms for analyzing the data based on these models. It exemplifies the view of applied mathematics as starting with a collection of problems from some area of science and then seeking the appropriate mathematics for clarifying the experimental data and the underlying processes of producing these data. An emphasis is placed on finding the mathematical and, where needed, computational tools needed to reach those goals, actively involving the reader in this process. Among other examples and problems, the following areas are treated: music as a realvalued function of continuous time, character recognition, the decomposition of an image into regions with distinct colors and textures, facial recognition, and scaling effects present in natural images caused by their statistical selfsimilarity.
Pattern Theory, pioneered by Ulf Grenander, is a distinctive approach to the analysis of all forms of real-world signals. At its core is the design of a large variety of probabilistic models whose samples reproduce the look and feel of the real signals, their patterns, and their variability. Bayesian statistical inference then allows you to apply these models in the analysis of new signals. This book treats the mathematical tools, the models themselves, and the computational algorithms for applying statistics to analyze six representative classes of signals of increasing complexity.
"Pattern Theory covers six classic attempts at modeling signals from the human and natural world: natural language (written), music, character recognition, texture modeling, face recognition, and natural scenes. These applications, appealing to students and researchers alike, include fourteen "crash courses" giving all the needed basics, exercises, and numerical simulations. Thus, it is a complete pedagogic tool at master or first-year graduate level. I endorse the publication of Pattern Theory, and will actually use it and recommend it to other researchers."---Jean-Michel Morel, CMLA
"This book is fascinating. It develops a statistic approach to finding the patterns in the signals generated by the world. The style is lucid. I'm reminded of Mumford's exposition of Theta functions and Abelian varieties in his Tata lectures. The exposition is thorough. The authors provide the necessary mathematical tools allowing scientists to pursue an exciting subject. I've been running a seminar at MIT entitled `New Opportunities for the Interactions of Mathematics and Other Disciplines' because I'm convinced that mathematics will move in surprising new directions. Pattern Theory, a decade's effort, is a prime example."---I. M. Singer, Institute Professor, MIT
"What singles out this outstanding book is an extremely original approach ... The authors are leaders in signal and image processing and this book is based on their innovative research work. The overall organization of the book is marvelous. It is a crescendo. The authors do not have any methodological prejudice. Reading this book is entering David Mumford's office and beginning a friendly and informal scientific discussion with Agnes and David. That is a good approximation to paradise."---Yves Meyer, Membre de l'Institut, Foreign honorary member of the American Academy of Arts and Sciences
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